Integrality conjecture for logarithmic HLV kernels

Let $\mathbb H_{u_1,u_2,\ldots,u_g}[X_1,X_2,\ldots,X_n;q,t,T]$ be the logarithmic HLV kernel, and let $\mathbb H_{u_1,u_2,\ldots,u_g,\lambda}(q,t)$ denote the coefficient of $\mathbb H$ in front of the monomial

$$\prod_{i=1}^n\prod_j X_{ij}^{\lambda^{(i)}_j}$$

for a tuple of partitions $\lambda=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(n)})$. Part 1. For every such tuple $\lambda$,

$$\mathbb H_{u_1,u_2,\ldots,u_g,\lambda}(q,t)\in\mathbb Z[q,t,u_1,\ldots,u_g,u_1^{-1},\ldots,u_g^{-1}].$$

This is the integrality part of the HLV-kernel conjectures, formulated in connection with character varieties and moduli spaces of Higgs bundles. The supplied text does not state whether this conjecture has been resolved.